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Automorphisms of Manifolds and Algebraic $$K$$-Theory: Part III
Michael S. Weiss, Mathematisches Institut, Universität Münster, Germany, and Bruce E. Williams, University of Notre Dame, Indiana
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Memoirs of the American Mathematical Society
2014; 110 pp; softcover
Volume: 231
ISBN-10: 1-4704-0981-X
ISBN-13: 978-1-4704-0981-4
List Price: US$71 Individual Members: US$42.60
Institutional Members: US\$56.80
Order Code: MEMO/231/1084

The structure space $$\mathcal{S}(M)$$ of a closed topological $$m$$-manifold $$M$$ classifies bundles whose fibers are closed $$m$$-manifolds equipped with a homotopy equivalence to $$M$$. The authors construct a highly connected map from $$\mathcal{S}(M)$$ to a concoction of algebraic $$L$$-theory and algebraic $$K$$-theory spaces associated with $$M$$. The construction refines the well-known surgery theoretic analysis of the block structure space of $$M$$ in terms of $$L$$-theory.

• Introduction
• Outline of proof
• Visible $$L$$-theory revisited
• The hyperquadratic $$L$$-theory of a point
• Excision and restriction in controlled $$L$$-theory
• Control and visible $$L$$-theory
• Control, stabilization and change of decoration
• Spherical fibrations and twisted duality
• Homotopy invariant characteristics and signatures
• Excisive characteristics and signatures
• Algebraic approximations to structure spaces: Set-up
• Algebraic approximations to structure spaces: Constructions
• Algebraic models for structure spaces: Proofs
• Appendix A. Homeomorphism groups of some stratified spaces
• Appendix B. Controlled homeomorphism groups
• Appendix C. $$K$$-theory of pairs and diagrams
• Appendix D. Corrections and elaborations
• Bibliography